Optimal Decay Estimate and Asymptotic Profile for Solutions to the Generalized Zakharov-Kuznetsov-Burgers Equation in 2D
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| Format: | Preprint |
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2023
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| _version_ | 1866913496403279872 |
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| author | Fukuda, Ikki Hirayama, Hiroyuki |
| author_facet | Fukuda, Ikki Hirayama, Hiroyuki |
| contents | We consider the Cauchy problem for the generalized Zakharov-Kuznetsov-Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative equations, which has a spatial anisotropic dissipative term $-μu_{xx}$. In this paper, we prove that the solution to this problem decays at the rate of $t^{-\frac{3}{4}}$ in the $L^{\infty}$-sense, provided that the initial data $u_{0}(x, y)$ satisfies $u_{0}\in L^{1}(\mathbb{R}^{2})$ and some appropriate regularity assumptions. Moreover, we investigate the more detailed large time behavior and obtain a lower bound of the $L^{\infty}$-norm of the solution. As a result, we prove that the given decay rate $t^{-\frac{3}{4}}$ of the solution to be optimal. Furthermore, combining the techniques used for the parabolic equations and for the Schr$\ddot{\mathrm{o}}$dinger equation, we derive the explicit asymptotic profile for the solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_12374 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal Decay Estimate and Asymptotic Profile for Solutions to the Generalized Zakharov-Kuznetsov-Burgers Equation in 2D Fukuda, Ikki Hirayama, Hiroyuki Analysis of PDEs 35B40, 35Q53 We consider the Cauchy problem for the generalized Zakharov-Kuznetsov-Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative equations, which has a spatial anisotropic dissipative term $-μu_{xx}$. In this paper, we prove that the solution to this problem decays at the rate of $t^{-\frac{3}{4}}$ in the $L^{\infty}$-sense, provided that the initial data $u_{0}(x, y)$ satisfies $u_{0}\in L^{1}(\mathbb{R}^{2})$ and some appropriate regularity assumptions. Moreover, we investigate the more detailed large time behavior and obtain a lower bound of the $L^{\infty}$-norm of the solution. As a result, we prove that the given decay rate $t^{-\frac{3}{4}}$ of the solution to be optimal. Furthermore, combining the techniques used for the parabolic equations and for the Schr$\ddot{\mathrm{o}}$dinger equation, we derive the explicit asymptotic profile for the solution. |
| title | Optimal Decay Estimate and Asymptotic Profile for Solutions to the Generalized Zakharov-Kuznetsov-Burgers Equation in 2D |
| topic | Analysis of PDEs 35B40, 35Q53 |
| url | https://arxiv.org/abs/2311.12374 |